reversed if backlash exists,
the pitch-curves are thrown out of contact, and, although the continuity
of the motion will not be interrupted, the velocity ratio will be
affected. If the motion is never to be reversed, the perfect law of the
velocity ratio due to the elliptical pitch-curve may be preserved by
reducing the thickness of the tooth, not equally on each side, as is
done in circular wheels, but wholly on the side not in action. But if
the machine must be capable of acting indifferently in both directions,
the reduction must be made on both sides of the tooth: evidently the
action will be slightly impaired, for which reason the backlash should
be reduced to a minimum. Precisely what _is_ the minimum is not so easy
to say, as it evidently depends much upon the excellence of the tools
and the skill of the workman. In many treatises on constructive
mechanism it is variously stated that the backlash should be from
one-fifteenth to one-eleventh of the pitch, which would seem to be an
ample allowance in reasonably good castings not intended to be finished,
and quite excessive if the teeth are to be cut; nor is it very obvious
that its amount should depend upon the pitch any more than upon the
precession of the equinoxes. On paper, at any rate, we may reduce it to
zero, and make the teeth and spaces equal in breadth, as shown in the
figure, the teeth being indicated by the double lines. Those upon the
portion L H are then laid off upon K I, after which these divisions are
transferred to the ellipse by the second of Prof. Rankine's
constructions, and we are then ready to draw the teeth.

The outlines of these, as of any other teeth upon pitch-curves which
roll together in the same plane, depend upon the general law that they
must be such as can be marked out upon the planes of the curves, as they
roll by a tracing-point, which is rigidly connected with and carried by
a third line, moving in rolling contact with both the pitch-curves. And
since under that condition the motion of this third line, relatively to
each of the others, is the same as though it rolled along each of them
separately while they remained fixed, the process of constructing the
generated curves becomes comparatively simple. For the describing line
we naturally select a circle, which, in order to fulfil the condition,
must be small enough to roll within the pitch ellipse; its diameter is
determined by the consideration that if it be equal to A P, the r